Flavor-dependent $U(1)$ extension inspired by lepton, baryon and color numbers

There is no reason why the gauge symmetry extension is family universal as in the standard model and the most well-motivated models, e.g. left-right symmetry and grand unification. Hence, we propose a simplest extension of the standard model – a flavor-dependent $U(1)$ gauge symmetry – and find the new physics insight. For this aim, the $U(1)$ charge, called $X$, is expressed as $X=xB+yL$ in which $x$ and $y$ are free parameters as functions of flavor index, e.g. for a flavor $i$ they take $x_i$ and $y_i$ respectively, where $B$ and $L$ denote normal baryon and lepton numbers. Imposing a relation involved by the color number $3$, i.e. $-x_{1,2,\cdots ,n}=x_{n+1,n+2,\cdots ,n+m}=3y_{1,2,\cdots ,n+m}\equiv 3z$, for arbitrarily nonzero $z$, we achieve a novel $U(1)$ theory with implied $X$-charge. This theory not only explains the origin of the number of observed fermion families but also offers a possible solution for both neutrino mass and dark matter, which differs from $B-L$ extension. Two typical models based on this idea are examined, yielding interesting results for flavor-changing neutral currents and particle colliders, besides those of neutrino mass and dark matter.